Paper · Jan 22, 2026

Four-dimensional Lorentzian algebraic Ricci solitons

Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey, Ramon Vazquez-Lorenzo

math.DGarXiv:2601.15730
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As on arXiv

Abstract

We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any connected and simply connected four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton.

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Submitted
Jan 22, 2026
Subjects
math.DG
Ricci solitons4-manifoldsHomogeneous spacesPhysics

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups (Rohollah Bakhshandeh-Chamazkoti, 2021)3Bakhshandeh- 2021 Ricci Solitons on a family of three dimensional Lorentzian Walker manifolds (A. Diatta, M. Ciss and A. S. Diallo, 2025)8Diatta 2025 Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group (Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim, 2025)5Calvaruso 2025 Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups (Youssef Ayad, 2026)2Ayad 2026 Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups (Youssef Ayad, 2026)1Ayad 2026 Lorentzian Algebraic Ricci Solitons on the Heisenberg Group (Youssef Ayad, 2026)4Ayad 2026 A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups (Fanqi Lin, 2026)7Lin 2026 Generalized Ricci Solitons on Heisenberg and Oscillator Algebras (Vicente Cortés, Shubham Dwivedi and Lorenz Jetter, 2026)6Cortés 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Physics, Homogeneous spaces · Shared terms “algebraic Ricci”, “Lorentzian”, “Lie groups”

  2. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “algebraic Ricci”, “dimensional Lie”, “four dimensional”

  3. Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

    Rohollah Bakhshandeh-Chamazkoti

    Topics 4-manifolds, Physics, Homogeneous spaces · Shared terms “invariant Lorentz”, “Lorentz metric”, “left invariant”

  4. Lorentzian Algebraic Ricci Solitons on the Heisenberg Group

    Youssef Ayad

    Topics Physics, Ricci solitons · Shared terms “Lorentzian algebraic”, “algebraic Ricci”, “left invariant”

  5. Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

    Giovanni Calvaruso, Lorenzo Pellegrino and Amirhesam Zaeim

    Topics 4-manifolds, Physics, Homogeneous spaces · Shared terms “Lorentzian”, “four dimensional”, “Lie groups”

  6. Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

    Vicente Cortés, Shubham Dwivedi and Lorenz Jetter

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “four dimensional”, “algebraic”, “left invariant”

  7. A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups

    Fanqi Lin

    Topics Physics, Homogeneous spaces, Ricci solitons · Shared terms “dimensional Lorentzian”, “left invariant”, “Lie groups”

  8. Ricci Solitons on a family of three dimensional Lorentzian Walker manifolds

    A. Diatta, M. Ciss and A. S. Diallo

    Topics Physics, Ricci solitons · Shared terms “dimensional Lorentzian”, “Lie”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.